A significant consideration which must be faced by financial institutions (and individual investors) is the potential risk of future losses which is inherent in a given financial position, such as a portfolio. There are various ways for measuring potential future risk which are used under different circumstances. One commonly accepted measure of risk is the value at risk (“VAR”) of a particular financial portfolio. The VAR of a portfolio indicates the portfolio's market risk at a given percentile. In other words, the VAR is the greatest possible loss that the institution may expect in the portfolio in question with a certain given degree of probability during a certain future period of time. For example, a VAR equal to the loss at the 99th percentile of risk indicates that there is only a 1% chance that the loss will be greater than the VAR during the time frame of interest.
Generally, financial institutions maintain a certain percentage of the VAR in reserve as a contingency to cover possible losses in the portfolio in a predetermined upcoming time period. It is important that the VAR estimate be accurate. If an estimate of the VAR is too low, there is a possibility that insufficient funds will be available to cover losses in a worst-case scenario. Overestimating the VAR is also undesirable because funds set aside to cover the VAR are not available for other uses.
To determine the VAR for a portfolio, one or more models which incorporate various risk factors are used to simulate the price of each instrument in the portfolio a large number of times using an appropriate model. The model characterizes the price of the instrument on the basis of one or more risk factors, which can be broadly considered to be a market factor which is derived from tradable instruments and which can be used to predict or simulate the changes in price of a given instrument. The risk factors used in a given model are dependent on the type of financial instrument at issue and the complexity of the model. Typical risk factors include implied volatilities, prices of underlying stocks, discount rates, loan rates, and foreign exchange rates. Simulation involves varying the value of the risk factors in a model and then using the model to calculate instrument prices in accordance with the selected risk factor values. The resulting price distributions are aggregated to produce a value distribution for the portfolio. The VAR for the portfolio is determined by analyzing this distribution.
A particular class of instrument which is simulated is an option. Unlike simple securities, the price of an option, and other derivative instruments, is dependant upon the price of the underlying asset price, the volatility of changes in the underlying asset price, and possibly changes in various other option parameters, such as the time for expiration. An option can be characterized according to its strike price and the date it expires and the volatility of the option price is related to both of these factors. Sensitivity of the option volatility to these effects are commonly referred to skew and term. Measures of the volatility for a set of options can be combined to produce a volatility surface. For example, FIG. 1 is a graph of the implied volatility surface for S&P 500 index options as a function of strike level and term to expiration on Sep. 27, 1995.
The volatility surface can be used to extract volatility values for a given option during simulation. The extracted volatility value is applied to an option pricing model which provides simulated option prices. These prices can be analyzed to make predictions about risk, such as the VAR of a portfolio containing options. The volatility surface is not static, but changes on a day-to-day basis. Thus, in order to make risk management decisions and for other purposes, changes in the volatility surface need to be simulated as well.
Various techniques can be used to simulate the volatility surface over time. In general financial simulations, two simulation techniques are conventionally used: parametric simulation and historical simulation and variations of these techniques can be applied to simulate volatilities.
In a parametric simulation, the change in value of a given factor is modeled according to a stochastic or random function responsive to a noise component ε is a noise component. During simulation, a suitable volatility surface can be used to extract a starting volatility value for the options to be simulated and this value then varied in accordance with randomly selected values of noise over the course of a simulation.
Although parametric simulation is flexible and permits the model parameters to be adjusted to be risk neutral, conventional techniques utilize a normal distribution for the random noise variations and must explicitly model probability distribution “fat-tails” which occur in real life in order to compensate for the lack of this feature in the normal distribution. In addition, cross-correlations between various factors must be expressly represented in a variance-covariance matrix. The correlations between factors can vary depending on the circumstances and detecting these variations and compensating is difficult and can greatly complicate the modeling process. Moreover, the computational cost of determining the cross-correlations grows quadratically with the number of factors making it difficult to process models with large numbers of factors.
An alternative to parametric simulation is historical simulation. In a historical simulation, a historical record of data is analyzed to determine the actual factor values and these values are then selected at random during simulation. This approach is extremely simple and can accurately capture cross-correlations, volatilities, and fat-tail event distributions. However, this method is limited because the statistical distribution of values is restricted to the specific historical sequence which occurred. In addition, historical data may be missing or non-existent, particularly for newly developed instruments or risk factors, and the historical simulation is generally not risk neutral.
Accordingly, there is a need to provide an improved method for simulating a volatility surface to determine volatility values during option pricing simulation.
It would be advantageous if such a method captured cross-correlations and fat-tails without requiring them to be specifically modeled and while retaining the advantageous of parametric modeling techniques.
It would also be advantageous if such a method could be extended to other multi-variant factors which are used in option pricing models.